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Date _________________ Item _______ 30-60-90 Triangle Discovery 1. Observe ΔPQR . Use patty paper to trace one of the angles, and then compare it to the other angles. Confirm that the triangle is equiangular. Use what you know about the sum of the angles in a triangle to determine the angle measures. ∠Q = ____ ∠R = ____ ∠P = ____ Record the measures for ∠P and ∠R on the diagram. Do not record the measure for ∠Q on the diagram. 2. Use a straightedge to draw an altitude starting at point Q. This altitude should intersect PR where you see a point. Label this point S. Use what you know about altitudes and the sum of the angles in a triangle to determine the angle measures. ∠QSR = ____ ∠SQR = ____ Record the measures for ∠QSR and ∠SQR on the diagram. 3. Use patty paper to trace ΔQRS and compare it to ΔQPS . What do you notice? 4. Focus on the full triangle again-- ΔPQR . Use patty paper to trace one of the sides, and then compare it to lengths of the other sides. Confirm that the triangle is equilateral. The triangle is shown on centimeter paper. Record the lengths of the following segments. RS = ____ How did you determine the measure of QR ? QR = ____ 5. Use the Pythagorean Theorem for ΔQRS to determine the exact length of QS . Show your work below, and then record the length of QS on the diagram. 6. Observe ΔTUV . Confirm that ΔTUV is equiangular and equilateral. Use a straightedge to draw an altitude starting at point T. This altitude should intersect UV where you see a point. Label this point W. Determine the following angle measures and exact lengths, and record them below and on the diagram. If you use the Pythagorean Theorem, show your work. ∠U = ____ TU = ____ ∠TWU = ____ UW = ____ ∠WTU = ____ TW = ____ 7. Observe ΔXYZ . Confirm that ΔXYZ is equiangular and equilateral. Use a straightedge to draw an altitude starting at point X. This altitude should intersect YZ where you see a point. Label this point A. Determine the following angle measures and exact lengths, and record them below and on the diagram. If you use the Pythagorean Theorem, show your work. ∠Y = ____ XY = ____ ∠XAY = ____ AY = ____ ∠AXY = ____ AX = ____ 8. Share your discuss your results with a partner. Look for patterns in the relationship of the sides. Try to complete the following conjecture. In a 30-60-90 triangle, if the leg opposite the 30-degree angle has length x, then the leg opposite the 60-degree angle has length ______ and the hypotenuse has length ______.